General Harmonic Numbers/Examples/Order 1/One Third

Example of General Harmonic Number

$\harm 1 {\dfrac 1 3} = 3 - \dfrac 3 2 \ln 3 - \dfrac \pi {2 \sqrt 3}$

where $\harm 1 {\dfrac 1 3}$ denotes the general harmonic number of order $1$ evaluated at $\dfrac 1 3$.

$\ds \harm 1 z$

$=$

$\ds \map \psi {z + 1} + \gamma$

$\ds \leadsto \ \ $

$\ds \harm 1 {\dfrac 1 3}$

$=$

$\ds \map \psi {\dfrac 1 3 + 1} + \gamma$

setting $z := \dfrac 1 3$

$\ds $

$=$

$\ds \map \psi {\dfrac 4 3} + \gamma$

$\ds $

$=$

$\ds \paren {-\gamma - \dfrac 3 2 \ln 3 - \dfrac \pi {2 \sqrt 3} + 3} + \gamma$

$\ds $

$=$

$\ds 3 - \dfrac 3 2 \ln 3 - \dfrac \pi {2 \sqrt 3}$

$\blacksquare$